Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

This tag is for questions typically taught in precalculus, as well as elementary algebra.

These topics include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomials, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

47234 questions
8
votes
5 answers

Prove this number fact

Prove that $x \neq 0,y \neq 0 \Rightarrow xy \neq 0$. Suppose $xy = 0$. Then $\frac{xy}{xy} = 1$. Can we say that $\frac{xy}{xy} = 0$ and hence $1 = 0$ which is a contradiction? I thought $\frac{0}{0}$ was undefined.
Numberguy
  • 83
8
votes
4 answers

If $x^3+\frac{1}{x^3}=18\sqrt{3}$ then to prove $x=\sqrt{3}+\sqrt{2}$

If $x^3+\frac{1}{x^3}=18\sqrt{3}$ then we have to prove $x=\sqrt{3}+\sqrt{2}$ The question would have been simple if it asked us to prove the other way round. We can multiply by $x^3$ and solve the quadratic to get $x^3$ but that would be…
rah4927
  • 3,864
8
votes
2 answers

Dividing one equation by another equation

This is from Higher Algebra by Hall and Knight, $u+v+\sqrt{uv}=39$...(1) $u^2+v^2+uv=741$...(2) we obtain by division $u+v-\sqrt{uv}=19$ I don't know how do you divide one equation by another equation, can someone pls explain.
Vikram
  • 5,580
8
votes
2 answers

Average of 3 consecutive odd numbers

The average of $3$ consecutive odd numbers is $14$ more than one third of the first of these numbers, what is the last of these numbers? $17/19/15/$data inadequate/none of these Let three consecutive odd numbers be $a-2,a,a+2$. Their average is…
aarbee
  • 8,246
8
votes
1 answer

Calculation of real values of $x$ in $\sqrt{4^x-6^x+9^x}+\sqrt{9^x-3^x+1}+\sqrt{4^x-2^x+1} = 2^x+3^x+1$

Calculate the real solutions $x\in\mathbb{R}$ to $$ \tag1\sqrt{4^x-6^x+9^x}+\sqrt{9^x-3^x+1}+\sqrt{4^x-2^x+1} = 2^x+3^x+1 $$ My Attempt: Let $2^x = a$ and $3^x = b$ . Then $(1)$ becomes $$ \sqrt{a^2-a\cdot b+b^2}+\sqrt{b^2-b+1}+\sqrt{a^2-a+1} =…
juantheron
  • 53,015
8
votes
2 answers

How to find real root of a function with $ \sin x, \cos x $?

Is there any way to find real root of the following equation by hand? Only need to count the number of zeros~ $$x^2-x \sin x- \cos x=0 $$ I know there's a rule for finding real roots for polynomial by counting the number of times that the sign of…
Lily
  • 361
8
votes
3 answers

Forget about the sin and cos functions, show that $(x-x^3/3!+x^5/5!-x^7/7!+...)^2+ (1-x^2/2!+x^4/4!-x^6/6!+...)^2=1$.

Forget about the $\sin$ and $\cos$ functions, are there possibly some brilliant way to show that $$\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots\right)^2+ \left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots\right)^2=1$$ ? I've…
JSCB
  • 13,456
  • 15
  • 59
  • 123
8
votes
3 answers

Proving a polynomial to be positive for all real values

I'm trying to prove the following theorem: $$\sum_{j=0}^{2n} (-x)^j >0\ \forall x \in \mathbb{R}, n \in \mathbb{N} $$ I have verified this theorem for $n=1$ (just a quadratic) and for $n=2$ (by simple factoring). However I'm stuck for higher values…
Aarav Gupta
  • 403
8
votes
3 answers

Solve the equation $\frac{\sqrt{4+x}}{2+\sqrt{4+x}}=\frac{\sqrt{4-x}}{2-\sqrt{4-x}}$

Solve the equation $$\dfrac{\sqrt{4+x}}{2+\sqrt{4+x}}=\dfrac{\sqrt{4-x}}{2-\sqrt{4-x}}$$ The domain is $4+x\ge0,4-x\ge0,2-\sqrt{4-x}\ne0$. Note that the LHS is always positive, so the roots must also satisfy: $A:2-\sqrt{4-x}>0$. Firstly, I decided…
kormoran
  • 2,963
8
votes
1 answer

Nature of roots of two quadratic expressions

Given $\,\left|px^2 +qx +r\right|\leqslant\left|Px^2 +Qx +R\right|\,$ for all real $x$ where $P,Q$ and $R$ are different from $p,q$ and $r$, I wish to find the relation between the roots of these quadratic expressions assuming both of them have real…
Rishi Shekher
  • 443
8
votes
3 answers

About rationalizing expressions

For example, rationalizing expressions like $$\frac{1}{\pm \sqrt{a} \pm \sqrt{b}}$$ Is straightforward. Moreover cases like $$\frac{1}{\pm \sqrt{a} \pm \sqrt{b} \pm \sqrt{c}}$$ and $$\frac{1}{\pm \sqrt{a} \pm \sqrt{b} \pm \sqrt{c} \pm…
chubakueno
  • 5,623
8
votes
3 answers

Is there a contradiction is this exercise?

The following exercise was a resolution to this problem Let $\displaystyle\frac{2x+5}{(x-3)(x-7)}=\frac{A}{(x-7)}+\frac{B}{(x-3)}\space \forall \space x \in \mathbb{R}$. Find the values for $A$ and $B$ The propose resolution was: In order to…
user24047
8
votes
4 answers

Square root of $8^3$

I'm only in intermediate algebra. I know that $\sqrt{8^3}$ is equal to $16\sqrt{2}$ but could you simply explain the process on how to get to that?
user79477
  • 133
8
votes
5 answers

Complete the square for $f(x) = 2x^2 + 4x - 6$

I'm studying for a math test. This is the question: $f(x) = 2x^2 + 4x - 6$. complete the square. This is how much I get out of the question: $$2x^2 + 4x - 6$$ $$2(x^2 + 2x - 3)$$ $$2(x^2 + 2x + 1^2 - 1^2 - 3)$$ $$2((x + 1)^2 - 4)$$ But I get stuck…
Ernes Filipovic
  • 81
8
votes
9 answers

If $x \neq 0,y \neq 0,$ then $x^2+xy+y^2$ is .....

I came across the following problem that says: If $x \neq 0,y \neq 0,$ then $x^2+xy+y^2$ is 1.Always positive 2.Always negative 3.zero 4.Sometimes positive and sometimes negative. I have to determine which of the aforementioned…
learner
  • 6,726
1 2 3
…
99 100